Simplify. Rewrite the expression in the form $x^n$. $x^3\cdot x^5=$
Solution: $\begin{aligned} x^3\cdot x^5&=x^{3+5} \\\\ &=x^{8} \end{aligned}$ This follows from the general rule $x^m\cdot x^n=x^{m+n}$. Note that the powers have the same base. We can also see this is correct by expanding the powers. $\begin{aligned} x^3\cdot x^5&=\underbrace{x\cdot x\cdot x}_\text{3 times}\cdot\underbrace{x\cdot x\cdot x\cdot x\cdot x}_\text{5 times} \\\\\\ &=\underbrace{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x}_\text{8 times} \\\\ &=x^{8} \end{aligned}$ In conclusion, $x^3\cdot x^5=x^{8}$.